## Question 416:

1

1. The null hypothesis is that there is no difference between groups. The probability the difference between these two means is due to chance alone is around .011, which is less than .05 so we'd reject the Null hypothesis and conclude that there is a difference between groups (I conducted the 2-sample t-test in Minitab and the output is below).
2. The t-test for a single sample compares a sample to a specified value. For example, is the mean of the sample greater than 30. A t-test for two independent groups compares the means for two groups and answers the questions if there is evidence the means of the groups are different than each other, whereas the 1-sample test just tests if the mean is different than a value.
A sketch of the two distributions is in the attached jpg file.

To compute the t-test by hand, we need to calculate the test statistic t*. The steps are outlined in the attached word-document and the calculations are available in the attached excel file.

If you use Mintab, it will default to assuming unequal variances and you'll get a slightly different p-value. You can see the output below.

### Minitab Output (Assumed Unequal Variances)

Two-Sample T-Test and CI

Sample N Mean StDev SE Mean

1 20 38.00 3.00 0.67

2 30 35.00 5.00 0.91

Difference = mu (1) - mu (2)

Estimate for difference: 3.00000

95% CI for difference: (0.72101, 5.27899)

T-Test of difference = 0 (vs not =): T-Value = 2.65 P-Value = 0.011 DF = 47